BSTJ 48: 3. March 1969: Frequency Sampling Filters - Hilbert Transformers and Resonators. (Bogner, R.E.)

THE BELL SYSTEM

TECHNICAL JOURNAL

Volume 48 March 1969 Number 3

Copyright © 1969, American Telephone and Telegraph Company

Frequency Sampling Filters —
Hilbert Transformers and Resonators

By R. E. BOGNER

(Manuscript received November 6, 1968)

We first briefly review the principles of frequency sampling filters.
We also shoiu that the "conventional" frequency sampling filter can be
modified simply to give an output which is the Hilbert transform of the
original output. Both the original and transformed outputs are made
available by the use of the simple complex number resonator described. The
relationship between this system and filtering by Fourier transforming
is shown.

I. INTRODUCTION

Frequency sampling filters are niters whose frequency responses are
synthesized as the sum of elemental frequency responses of the form
(Fig. la) 1

si n \rr(f - />)//.] -r-rrr , 4 sin tt(/ + /*)//„ -,- ir/T (1)

where

V k (f) is the transfer function of the fcth response;

A k is a constant multiplier, the value of the amplitude response at

frequency /*•;
/ is frequency in hertz ;
f k is the fcth sampling frequency = kf ;

501

502

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1909

A k

-^=K>-

DT

DT

(a)

Fig. 1 — (a) Elemental frequency response contribution ; (b) Elemental time
response contribution.

/ is the frequency interval between samples, that is, /„ = /k+i— /*,

f = 1/DT, D = delay in samples;
t is the group delay, a constant for all the responses.

Because of the constant group delay, the amplitude versus frequency
response, IV (/) I, of the sum is given by

™ I ■ ? A t^l§t

+

sin *<j + U)/j

?■■••

(2)

By choice of the A k , suitable amplitude responses for many applica-
tions may be specified. These will be bandlimited functions of fre-
quency.

The elemental time responses, v k (t) (Fig. lb) are convenient to
realize by digital methods. They are truncated cosine waves.

Figure 2 shows a comb filter, whose impulses occur DT seconds
apart, followed by a resonator, whose impulse response is a cosine
wave of frequency an integral multiple of 1/DT. The overall impulse
response is the sum of the cosine responses to the two impulses; this
is zero before the positive impulse, a cosine from then until DT sec-
onds later, and thereafter zero, when the two cosines cancel.

A complete frequency sampling filter is shown in the left of Fig. 3.
Usually the resonators have been programmed as conventional second

COMB
FILTER

RESONATOR

~»| DT «--

WVA

■ 27Tkt

DT

Fig. 2 — Comb filter followed by cosine resonator.

FREQUENCY SAMPLING FILTERS

A

RESONATORS

503

Fig. 3 — Frequency sampling filter, followed by Hilbert transformer.

order systems, with slight damping to ensure stability under condi-
tions of error in the resonator coefficients.

II. USE AS HILBERT TRANSFORMER

A frequency sampling filter may be readily adapted to give an out-
put which is the Hilbert transform of that of the filter described above.
Consider the sampling filter (Fig. 3) followed by a Hilbert trans-
former, h(t). This is equivalent to the system of Fig. 4, where the one
Hilbert transformer has been replaced by one at the output of each
elemental filter. Now, in the original frequency sampling filter, the
fcth resonator has an impulse response, for time sampled systems

g h (nT) = cos <a k (nT), n = 0, 1, 2,

where T is the sampling interval. The Hilbert transformed version of
this is approximately

§ t (nT) = sin u k (nT).

The approximation is discussed in Appendix A. Thus to make a system
equivalent to the original frequency sampling filter plus Hilbert trans-
former, we need only replace the resonators by ones with impulse re-
sponses sin 03 k t. This could be done by use of modified second order delay
resonators; but the system of Fig. 5 is more convenient programwise
and is helpful conceptually. This system has the z transform system
function

jm - G (z) -

U(z) ~ ° W ~ 1 - z~ l exp [(a + jo)T]
and corresponding impulse response

g(nT) = e"V r ,n = 0, 1, • • • .

(3)

(4)

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1969

HILBERT

OUT

Fig. 4 — Frequency sampling filter with separate Hilbert transformers.

For a = 0, the real and imaginaiy parts are cos <anT and sin tatiT. A
small negative value of a would be used for stability.

The frequency sampling filter then has the form of Fig. 4, with each
channel containing one complex number resonator instead of the res-
onator plus Hilbert transformer. The output at each sampling time is
a complex number, whose real part corresponds to the output of a
conventional frequency sampling filter, and whose imaginary part is
an approximation to the Hilbert transform of the real part.

In Appendix A, the analysis of the approximation results in the
following observations:

(i) The Hilbert transformer cannot handle signals with frequencies
tending to zero.

(ii) For signals with low-frequency components, care is necessary
in specifying the frequency samples to ensure that the negative- fre-
quency tail of the positive- frequency response component is of small
amplitude.

(in) The errors are in the amplitude and not phase characteristics.

The system is capable of filtering a complex input, u + jv without
modification of the resonators.

III. RELATION TO DISCRETE FOURIER TRANSFORM

Consider a = 0. The response of the fcth resonator at time nT,
n = 0, 1, 2, • • • , to a unit pulse at time mT is exp [jw k (n — m)T].
Hence the response at time nT to a signal s(mT), m = • • • , —1,0,
1, 2, • • ■ is:

x k (nT) + jy k (nT) = £ s(mT) exp \p, t (n - m)T]

= exp(jco k nT) £ s(mT) exp (- jo: k mT) . (5)

FREQUENCY SAMPLING FILTERS

+ (a+jw)T

.505

w(t)=i(t)+jy(t)

OUT

Fig. 5 — Complex number resonator.

When the comb filter precedes the resonator, the effect of its nega-
tive impulse, occurring DT seconds after the positive impulse is to
add the second term of (6) :

x k (nT) + jy k {nT) = exp (ju k nT) £ s(mT) exp (-ju k mT)

m=--oo

- exp (ju k nT) £ s(m - D)T exp (-p> k mT)

m = — oo

= exp (jw 4 nr)[ £ «(wT) exp (-jo> k mT)

n-D I

- £ 8{mT) exp (-ju k mT) exp (-jw t DT) • (G)

HI = - CO -I

But DT is an integral multiple of the period 2ir/<a k as mentioned in
Section I; thus exp (— j<a k DT) = 1. Hence

x k (nT) + jy k {nT) = exp (p, k nT) £ s(roT) exp (-p> t mT). (7)

This expression may be recognized as an oscillation exp (jco k nT) whose
coefficient is the value at frequency «* of the Discrete Fourier Transform
(DFT) of s(mT), computed over the last D samples. The output of the
frequency sampling filter, taking into account the weights A k , is

x(nT) + jy(nT) = £ A k [x k (nT) + jy k (nT)]
k

£ exp fa k nT)A k £ s(mT) exp (-ja> k mT) . (8)

i m-n-D + l

This is the Fourier synthesis (inverse DFT) of the frequency function

A k £ s(mT) exp (-ju k mT), k = 1, 2, • • • ,

(9)

which may be regarded as the product of the running DFT of s(mT)
and a DFT whose values at frequencies u k are the A k .

506 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1969

Frequency sampling filtering is thus equivalent to filtering by Fou-
rier transforming, multiplying by a filter frequency function, and in-
verse transforming.

The filter frequency function (A k , k = 1, 2, . . .) has, so far, been
considered real. There is no reason why the A k should not be com-
plex, permitting the filter to have an arbitrary phase characteristic.
The complex values of the A k may be specified in cartesian or polar
form, the latter being more convenient for amplitude-phase specifica-
tion.

Another way of looking at the resonator output is obtained by re-
arranging (7) :

x k (nT) + jy k (nT) = £ s[(m + n)T] exp (-jco k mT). (10)

m— (D-l)

This may be recognized as the DFT of the last D values of s {mT) ,
shifted in time so that the latest occurs at time mT = 0.

IV. CONCLUSION

The use of complex number resonators in a frequency sampling
filter provides a Hilbert transformed output as well as the conven-
tional filtered output. The system can readily accept a complex time
function as input, and has a very simple flow chart. The output is
equivalent to that obtained by the use of Fourier transforms to per-
form filtering in the frequency domain.

A sampling filter subroutine using the ideas presented has been
written in Fortran IV. It has been used for filtering and Hilbert
transforming speech signals in a number of tasks.

V. ACKNOWLEDGMENT

Thanks are due to C. H. Coker, L. R. Rabiner and R. W. Schafer
for many helpful discussions. A recursive generation of the DFT is
given in Ref. 2.

APPENDIX A

Errors in the Hilbert Transjormer

A cosine wave, truncated in time, is the basis of the frequency sam-
pling filters. A correspondingly truncated sine wave has been used
as an approximation to the Hilbert transform of the cosine. The
errors in this approximation will be analyzed by comparing the

FREQUENCY SAMPLING FILTERS

507

Fourier transform of the truncated sine wave with that of the true
Hilbert transform of the cosine. The analysis is for continuous (that
is, nonsampled) sines and cosines.

The truncated cosine response is taken to be

h e (t) = cos

2rNt
T '

= 0,
The F transform of h c {t) is

T T

—- < t < —
2 " - 2

elsewhere.

T

HAD = 2

inxT^-f) sin»r(F + ^)

*T[i -|

-AH-fi

(ID

(12)

= H el (f) + H e2 (J), respectively.
HAD ma Y ue separated further into main responses and "tails"
(Fig. 6) :

H c {j) = H cX Ai) + #«!-(/) + H c2 AD + H c2 (J) (13)

where

H ct+ =H cl , / > 0; I -^f 1 , / - 0; 0, / <
// cl _ =0, / > 0; ^^ , / = 0; H cl , / <

H c2+ = H c2> / > 0; ^ , / = 0; 0, / <

ff e2 - = 0, / > 0; ^^ , / = o; H c2 , / < 0.

The F transform of the Hilbert transform [fc(0] of A e (<) is then

#.0) = -;'sgn(/)H e (/)

= -jff el+ (/) + #/„_(/) - jH c2 A1) + jff.2-0).
The truncated sine response is taken to be

h,(t) = sin — yr-

T T

2 - - 2

(14)
(15)

= 0,

elsewhere.

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1969

FREQUENCY

Fig. 6 — Components of elemental frequency response.
The F transform of h„(t) is

T
H.(1) = §

sin icT\j + j)

which by comparison with (11), (12), (13) is seen to be
H.(f) = -jH ci (j) + jH e2 (f)

= -jH el+ (i) - jH ei _(f) + jH c2+ (f) + jH t2 -(f).
Then from (15) and (17):

H.(j) = H c (j) - 2jH el .(f) + 2jH c2+ (j).

(16)

(17)

(18)

The error in approximating & c (j) by H.(j) is thus attributable to the
tails #„!_(/) and H e2+ (j), which are small for N ^> 1. From the defini-
tions (11), (12), (13), it follows that these tails are related:

#,!-(-/) = #.,+ (/).

(19)

In a complete frequency sampling filter, the transforms corresponding
to all the time responses are to be added. Errors in the "Hilbert trans-
formed" output, y, as compared with the straight filtered output, x,
are determined by the resultant tails; these tails may be of small
amplitude if suitable values are chosen for the frequency samples.
Just what criterion of smallness should be applied depends on the
application. Some general observations may be made, however:

FREQUENCY SAMPLING FILTERS 509

(i) The Hilbert transformer cannot be useful to zero frequency
because a zero frequency sample has tails equal to the main responses,
and would thus contribute gross errors. This is of course consistent
with the infinite duration of the impulse response {1/t) of a true
Hilbert transformer.

(ii) To transform signals with low frequency components, many
frequency samples may be required to provide the sharp and con-
tinued cutoff required for tail suppression.

(Hi) Since # cl _(-/) = H c2+ (f), it follows from (18) that the
errors, associated with H cl -{-f) and H c2+ (f) are directly in or out
of phase with the relevant main responses. The error in the Hilbert
transform is thus an amplitude and not a phase error. This result is
also consistent with the observation that the approximate Hilbert
transformed response to an impulse is truly odd.

APPENDIX B

Relationship between Complex Number Resonator and Conventional
Second Order Resonator

While the formal transform relation between (3) and (4) is readily
shown, it is satisfying to explain how the seemingly first order delay
system can produce an oscillatory response. The system of Fig. 5 is
described by the equation

x(mT) + jy(mT) = u{mT) + e (a+ ' u) T [x(m - 1)T + jy(m - 1)T] (20)

When a pulse it(0) = 1, with zero before and after is applied, the first
response is

x(0) + MO) = 1 + jO

The next response is simply the first response multiplied by e (a + "* >7

x(lT)+jy(lT) =e (o + ' u,r (l+j0);

there is a similar multiplication at each subsequent sampling instant,
yielding the impulse response

x(nT) + jy{nT) = e n{a + i " T) , n = 0, 1, 2 • ■ ■ , (21)

equivalent to (4) .

The complex number resonator may be shown to contain a second
order delay feedback, making its oscillatory response consistent with
that of the more conventional second-order systems. Its equation (20)

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1969

Fig. 7 — Expanded flow chart for complex number resonator.

may be examined by equating separately real and imaginary parts:

x(mT) = u(mT) + (e aT cos uT)x[{m - l)T]

- (e a T sin ioT)y[(m - 1)T] (22)
y(mT) = (e aT sin a>T)x[(m - 1)T] + (e aT cos <*T)y[{m - \)T] (23)

Equations (22) and (23) may be represented by the flow chart of
Fig. 7. There is, in fact, a path of delay two sampling intervals from
the real output x, via y, the imaginary part of the output, back to
x. Thus, y could be considered to provide the necessary memory for
the second delay.

One aesthetically pleasing feature of the representation (Fig. 7) is
the symmetry. If a complex input, u + jv were to be filtered, then v
would be found to be applied to the lower summer.

REFERENCES

1. Rader, C. M. and Gold, B., "Digital Filter Design Techniques in the Fre-

quency Domain," Proc. IEEE, 55, No. 2 (February 1967), pp. 149-171.

2. Halberstein, J. H., "Recursive, Complex Fourier Analysis for Real-Time

Applications," Proc. IEEE Letters, 54, No. 6 (June 1966), p. 903.